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207 Cards in this Set
- Front
- Back
prime factors
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1 and number
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GCF
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greatest common factor
if nothing else, 1 |
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LCM
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least common multiple
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Reflexive axiom of equality
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a=a
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Symmetric axiom of equality
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if a=b, then b=a
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Transitive axiom of equality
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a=b, b=c, then a=c
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Closure operations
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a, b a+b, axb
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Associative operations
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(a+b)+c ===a+(b+)
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Identity operation (+/x)
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+ 0, x 1
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Inverse operation (+/x)
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-a, 1/a
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Commutative Axiom of +/x
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a+b=b+a
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Distributive operation of multiplication over addition
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a(b+c)= ab+ac
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Substitution property
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if a=b, then can use either
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Transitive vs Substitution
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T showing equality between two previously unrelated things
S replacing parts known to be equal. |
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Complex number
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-i, sqrt-1
Complex vs Real (rational + irrational) |
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Real numbers
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Rational + Irrational
Rational: Integers, Whole, Naturals Irrational: no fraction or repeating dec. |
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Rational
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any number st a/b,
fractions of whole numbers, positive and negative, no zero in denominator, repeating decimal |
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Irrational
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no fraction or repeating dec
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Integers
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Naturals (no fractions)
Whole (zero) Integers (-) |
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Natural
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counting numbers
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Whole
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zero
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integers
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negative
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index
radical sign radicand |
n
sqrt radicand^m = base^m/n |
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sqrt a x sqrt b
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sqrt ab
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a^p x a^q
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a^p+q
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a^p/a^q
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a^p-q
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(a^p)^q
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a^pq
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(ab)^p
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a^p b^p
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(a/b)^p
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a^p/b^p
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Arithmetic Series
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a(n) = a + (n-1)d
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Arithmetic sequence = sum
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S(n) = .5 n (a(1) + a(n))
n > 0 only find difference, recursive, linear function, exponential function y=ab^x (i.e. a(n)=10n=10,20,30, 40.... |
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Geometric series
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a(n) = a(1) r^n-1
n>0 a(1) = 2x5^(1-1) = 2 x 1 |
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Geometric Sequence = sum
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S(n) = a(1) ((1-r^n)/1-r)
n>0 |
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to solve xyz
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must have three equations
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first degree equation
slope intercept form from points on a line |
ax + by = c
y = mx +b , m=change y/change x Y-ya = m(X-xa) |
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Inequality word problems
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find examples in c8
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composition function
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(fog)x or f(g(x))
their domain requires both functions be defined (nonneg squares and no zero in denominator) ie sqrt x+2 x>2 |
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solve for xyz
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homogenous system has one trivial solution
if # variables greater or equal to number of equations, then non-trivial solutions. Use matrices and substitution. |
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Factor
x^2 + 10x - 24 = 0 |
(x+12) (x-2) = 0
x= -12, 2 |
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Complete the square
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move the nonvariable number to the other side and figuring out what number will complete factor st it is squared, add to both sides, take sq root.
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Quadratic Formula
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-b +/- sqrt (b^2 - 4ac)
--------------------------------- 2a Discriminant tells roots (0=one real, >0 2 unequal, <0 complex) |
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Quadratic from roots
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add roots = sum; multiply = product
x^2 + (opp of the sum)x + product = 0 eliminate fractions with common denom |
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roots of quadratic equation
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x intercepts
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composite factors
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greater than two primes
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divisibility for 3s and 4s
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sum multiples of three
last 2 digits 4 last 3 digits 8 |
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scientific notation operations
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add/subtract: maintain same exponent
multiply add exponents 120 add exp to left (1.20 x 10^2) divide subtract exponents .012 subtract exp to right (1.20 x10^-2) |
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what counts as significant digits?
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nonzero number
zeros between nonzeros final zeros (right of the decimal place) |
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mapping vs function
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vertical line test
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angle of exterior linear pair
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sum of opposite internal angles
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angle of central angle
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length of the minor arc
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inscribed angle
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half the intercepted arc measure
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vertical angles (where two chords intersect)
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half the sum of intercepted arcs
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tangent chord angle
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half the intercepted arc
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exterior vertice of a circle
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half the difference between the two arcs it inscribes
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relationship between segments of intersecting chords x and y
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x1x2 = y1y2
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relationship between intersecting tangent chords x and y
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x = y
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relationship between intersecting tangent and segments of secant chord x (tan) and y (sec)
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x^2 = y1 y2
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Perimeter of a regular polygon of n sides
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n x side length
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Area of a regular polygon of n sides
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half the sum of apothem x perimeter
to find apothem: if hexagon, 30 60 90 rules (radius=side length; a =halfside x sqrt3) |
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apothem of a polygon
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use angle relationships, if 345 triangle or if 30 60 90: radius = side length.
apothem = half radius x sqrt 3 (radius)^2 = (half radius)^2 + (halfradius x sqrt 3)^2 |
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Finding the domain of a function
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anything not negative in denominator and any square not negative
undefined and complex |
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To find max values of coins
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remember 5n +10d= ____
graph, use x and y intercepts for linear equalities |
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30 60 90
45 45 90 |
x, x sqrt3, 2x
x, x, x sqrt2 |
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median
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count data points, divide number in half. (average if even data set)
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mean
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sum data/number of data points
One problem with using the mean, is that it often does not depict the typical outcome. Outlier will skew the mean strongly affecting the outcome. The median is the middle score. If we have an even number of events we take the average of the two middles. Median better for typical value. It is often used for income and home prices. |
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mode
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most frequent data point
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range
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high to low data points
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percentiles
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data/100
one hundred equal parts of data |
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stanines
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linked to standard deviation: values to probability (y axis); steep curve is tight cluster around mean; low slope large standard deviation.
mean at the middle (highest part) of the bell curve, negative and positive standard deviations from the mean x axis |
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quartiles
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using medians to divide the data set into four subsets.
Quartiles are the medians of each of the four subsets (Q1-4) |
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variance
standard deviation |
Calculate the mean, x.
Write a table that subtracts the mean from each observed value. Square each of the differences. Add this column. Divide by n -1 where n is the number of items in the sample This is the variance. Take the sqrt(variance) = standard deviation from the mean To get the standard deviation we take the square root of the variance. |
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Why is standard deviation useful?
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The mean may either be closely spaced or spread out; it may be a good approximation of the data or a dangerously poor one. Examples: insulin level after a drug trial.
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radians to angles
30 45 60 90 |
30 = pi/6
45 = pi/4 60 = pi/3 90 = pi/2 |
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sinx
cscx |
OPP/HYP = SINX
HYP/OPP = CSCX |
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cosx
secx |
ADJ/HYP = COSX
HYP/ADJ = SECX |
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tanx
cotx |
OPP/ADJ = TANX
ADJ/OPP = COTX |
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direct variance in functions
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y= cx or y=cx^2
proportional equation |
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indirect variance in functions
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xy = c or y=c/x or y= c/x^2
inverse variation example: speed vs driving time |
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Absolute value equation
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y=m(x-h)+k
(h,k) max or min point + or - m = slopes solve for both equations (0, 1, 2) resubstitute finding null sets |
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multiply an inequality by -1
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reverses the inequality
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factoring cubes x^3 + y^3
acronym equation |
SDP== same different plus
x^3 + y^3 =(x+y) (x^2-xy+y^2) x^3 -- y^3 =(x--y) (x^2+xy+y^2) |
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matrices
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easy
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synthetic division
(x^4-3x+5)/ (x-4) |
x-4---> 4
PC polynomial coefficients 1 0 0 -3 5 bring down first PC (1) multiply by 4 replace in second column (under 0) add column repeat multiplication/replacement answer x^3+4x^2+16x+61+(249/(x-4)) |
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geometric postulates of congruency
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SSS
SAS (included angle) ASA (included side) NO ASS! |
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postulates
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accepted as true without proof
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geometric theorems derived from ASA
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AAS (two angles and noninclusive side)
HL if hypotenuse and leg of two right triangles are congruent, congruent. |
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Theorem
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derived from postulate
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Undefined terms
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point, line, plane
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defined geometric terms
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ray-- one direction from end point
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Property of addition
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if a=b and c=d, then a+c=b+d
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Property of Subtraction
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if a=b and c=d, then a-c=b-d
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Property of Multiplication
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if a=b then ac=bc
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Property of division
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if c does not equal zero and a=b,
then a/c=b/c |
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Reflexive Property
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a=a
what use is reflexive? |
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Symmetric property
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If a=b, then b=a
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Transitive property
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If a=b, b=c, then a=c
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Distributive Property of mult over addition
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a(b+c) = ab+ac
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Substitution property
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If a=b, b may be substituted for a
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distance between two points
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abs value (a-b or b-a)
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Induction
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examples patterns
sometimes true |
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Deduction
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from postulates->conclusion
always true |
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Internal angles of a polygon
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((n-2)180)/n
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External angles of a polygon
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360/n
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Isosceles Angle Theorem
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If two sides (or two angles) are equal, the angles (or sides) are equal.
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Parallel line theorem
Three examples |
corresponding angles congruent
alternate interior angles congruent alternate outside angles congruent |
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Area of a Circle
Volume of a Sphere Surface area of a sphere |
pi r^2
4/3 pi r^3 4 pi r^2 |
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Area Trapezoid
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h/2 (b1+b2)
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Area square
Area triangle Area parallelogram |
Asquare = lw
Atriangle = 1/2 base height Aparallelogram = base height |
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Volume cylinder
Surface Area |
V cylinder = pi r^2 h
SA cyl = 2 pi r h + 2 pi r^2 |
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Volume cone
Surface area |
Vcone = pi r^2 height/ 3
SA cone = pi r sqrt(r^2 + h^2) + pi r^2 |
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slant height of cone
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sprt(r^2 + h^2)
application of pythagoras theorem |
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Describe this parabola
y=a(x-h)^2 +k y=-3x^2 + 6x-1 |
vertex (h,k)
sym about the y axis x^2 +a indicates opens upward Complete the square y=-3(x^2-2x+1) -1 +3 y=-3(x-1)^2 +2 |
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Describe this parabola
x=-a(y+k)^2 -h |
vertex (-h,-k)
sym about the x axis (y^2) -a indicates opens downward complete the square |
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Describe this ellipse
(x-h)^2 //a^2 + (y-k)^2//b^2 =1 |
complete the square for y and x polynomials separately
center (h,k) x and y intercepts by solving for x=0 and y=0 length of major axis is 2a, minor axis is 2b c^2=a^2-b^2 foci of two circles that form the ellipse are either (h+/-c,0) or (0,h+/-c) |
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Describe an ellipse where b>a
a>b |
symmetrical about the y axis
symmetrical about the x axis |
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Describe this hyperbola
(x-h)^2//a^2 - (y-k)^2//b^2 = 1 |
-y^2 about y axis
and equation for asymptote is y= +/-b/a(x-h) +k -x^2 about x axis and equation for asymptote is y=+/-a/b(x-h)+k foci (h+/c, k) or (h,k+/-c) where c^2=a^2+b^2 (not pythag) vertices are (h+/-a, k) or (h, K+/-a) |
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Describe this circle
x^2 + y^2 = 9 |
if (x-h) or (y-k), the center is on (h,k) (ignore the negative sign)
here, center is (0,0) and the x and y intercepts are at sqrt9=+/-3 If |
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objective (No of queens/Total cards)
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P(x) =
Total number (outcomes)/ Total Possible 4/52 P(x) is always between 0 and 1, and the sum of all probabilities in a set is equal to 1 |
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P(heads) + P(tails)
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1/2 + 1/2 = 1
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mutually exclusive events
P(spade) + P(clubs) |
13/52 + 13/52 =26/52 =0.5
x or y |
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non-mutually exclusive
P(queen) + P(spade) |
4/52 + 13/52 - 1/52 = 16/32
x + y - P(x+y) |
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Independent Events
3 heads on 3 tosses |
1/2 x 1/2 x 1/2 = 1/8
P(x) x P(y) |
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Dependent Events
6 red, 4 green, 5 purple P(red) 1st draw and P(purple) on second draw |
P(red1)= 6/15
P(purple2)= 5/14 P(both) = 6/15 x 5/14 = 30/210 = 1/7 |
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Define Odds
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favorable outcomes//
unfavorable outcomes If odds against, then unfavorable//fav |
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Odds a head will turn up if three coins are tossed
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plot out the combinations, count favorable/unfavorable
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Combinations Problem
How many possible combinations of heads from five tosses? |
Pascal's Triangle: (1s are row zeros) fifth line summarizes total combinations of possible heads from 5 tosses (1,5, 10, 10, 5,1)
The full formula is =(n!/((n-r)!*r!)) |
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"how many different ways can you choose two objects from a set of three objects?"
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Using Pascals Triangle
1 11 121 13(3)1 = 3 ways. (place 2) If 5 objects, 14641 15(10) 10 5 1 = 10 ways. |
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permutations
10 objects, 4 selections without repeating, order is not important (ie. CACA is not the same as ACAC) |
10!/(10-4)! = 5040
More permutations than combinations |
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Combinations
10 objects, 4 selections without repeating, order is important (ie CACA equals AACC and ACAC ...) |
10!/[4!(10-4)!] = Permutation/4!
essentially dividing permutation by the 4 possible selection factorial |
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unit circle
pi/6=30 find sinx, cosx, tanx |
sinx = 1/2
cosx=sqrt3/2 tanx=1/sqrt3 |
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unit circle
pi/2 =90 find sinx, cosx, tanx |
sinx=1/1
cosx=0/1 tanx=1/0 or =/- infinity |
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Trig Identities
sin^2x + _____ = 1 |
cos^2x
sin^2x +cos^2x = 1 |
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Trig Identities
1 + _____ = sec^2x |
tan^2x
1 + tan^2x = sec^1x divide basic identity by cos^2x; solve. |
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Trig Identities
1 + _____ = csc^2x |
cot^2x
1 + cot^2x = csc^x divide basic identity by sin^2x |
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Law of Sines
Triangle with side a opposite Angle A |
sinA/a = sinB/b = sinC/c
The Law of Sines is used for solving triangles SSA (two sides and an angle opposite one of them) and AAS (or ASA) (two angles and any side). |
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Law of Cosines
assuming SAS (aCb) and want opposite c |
c^2 = a^2 +b^2 - 2ab cosC
very useful for finding third side length if you know other two and its opposite angle measure. |
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Derivatives of trig functions
d/dx(cosx) d/dx (sinx) |
-sinx
cosx |
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d/dx (tanx)
d/dx (cotx) |
quotient rule:
d/dx(sinx/cosx) = cosx (cosx) - sinx (-sinx)/cosx^2= cosx^2 -(-sinx^2)/cosx^2= 1/cosx^2= sec^2 d/dx(cotx) = -cscx^2 |
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Product Rule
d/dx (f(x) x g(x)) |
= f d/dx(g(x)) + d/dx(f(x) g(x)
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Quotient Rule
d/dx(f(x)/g(x)) |
g(x) d/dx(f(x)) - f(x) d/dx(g(x))// g(x)^2
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exp derivatives:
d/dx (e^x) |
d/dx (e^x) = e^x * d/dx(x) = 1 * e^x
The derivative of the exponential function is the exponential function |
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Three rules of logarithms
ln x*y ln x/y ln x^n ln1 |
ln x*y = ln x + ln y
ln x/y = lnx - lny ln x^n = n*lnx ln1 = 0 x intercept is zero; translation if ln(x-2) to +2 thus x intercept is at 3 and vertical asymptote is at +2 |
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log base 3 of 1/9
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log base 3 (1/3^2)= log base 3 (3^-2) =
-2 *log base 3 (3) = -2 *1 = -2 |
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b^n = x
e.g. 8^2 = 64 |
log base b (x) = n
2 = log base 8 (64) |
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log base b (b^x) =__
What exponent on the right will raise the base (b) to produce b^x |
log base b (b^x) = x
or b^x = (b^x) x -- on the right -- is the exponent to which the base b must be raised to produce bx. |
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Exponential functions
y=b^x y=b^0= |
y intercept is +1 thus b^0 = 1
why? |
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b^(logbaseb(x)) = _______
log base b (b^x) = _________ |
The inverse of any exponential function is a logarithmic function. For, in any base b:
b^(logbaseb(x)) = x (logbx is the exponent to which b must be raised to produce x.) log base b (b^x) = x composite inverse functions =x |
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f(g(x)) = ln e^x = ___
g(f(x)) = e^ln x =__________. |
x
lnx or log base e (x) is the inverse function of e^x: e^x y intercept 1 and lnx x intercept 1 about y=x |
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Solve this equation for x :
5^(x + 1) = 625 |
Solution. To "release" x + 1 from the exponent, take the inverse function -- the logarithm with base 5 -- of both sides.
logbase5(5^x + 1) = logbase5(625) x + 1 = logbase5(625) x + 1 = 4 x = 3. |
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Solve for x:
log2x + log2(x + 2) = 3. remember the domain of log base 2 x |
log2[x(x + 2)] = 3.
If we now let each side be the exponent with base 2, then x(x + 2) = 23 = 8. x² + 2x − 8 = 0 (x − 2)(x + 4) = 0 x = 2 or −4. See Skill in Algebra, Lesson 37. We must reject the solution x = − 4, however, because the negative number −4 is not in the domain of log2x. |
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Chain Rule
d/dx(x^3) d/dx(2/(x + x^2)3=d/dx 2(x+x^2)^-3 |
3x^2
-6(2x+1)//(x+x^2)^4 |
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Limits
lim(2x+1) = _____ ->2 lim (x^2-9)/(x+1) = ____ jx->3 |
=5 substitution
=0 if substitution results in inf or zero, use L'Hospital's rule: take the derivative of the equation (no rules) and plug in the limit again. |
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lne = _____
d/dx b^x =_________ d/dx 4^2x+5=_______ d/dx 2x(4^x)=________ |
ln e = 1
b^xlnb 2ln4(4^2x+5) =d/dx(b^u) =b^u*lnb du/dx (The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity.) 2(4^x) + 2x(4^x)ln4 (use this with product rule) |
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d/dx log base b (x) = 1/x*ln(b)
d/dx ln x = d/dx log base e (x) = 1/x |
The derivative of the natural logarithm of a quantity is the reciprocal of that quantity, times the derivative of that quantity.
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d/dx sin u =
d/dx cos u = |
cosx * du/dx
-sin x * du/dx |
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d/dx e^x=__________
d/dx e^2x+4 =_______ |
e^x
2e^2x+4 The derivative of e raised to a quantity is e raised to that quantity, times the derivative of that quantity. |
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d/dx lnx = ________
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1/x
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Derivative
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instantaneous rate of change
equation which describes the slope of the tangent at any point on a curve plug in value to find the slope or rate |
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Why is d/dx e^x = e^x
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e^x is the only function which grows at a rate of change equal to itself
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Quotient Rule
f(x) = g(x)//h(x) |
h(x) g'(x) - g(x) h'(x)//h(x)^2
straight bottom goes first and subtract it's differential |
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Differentiate
y=2^3x+1 * ln (5x-11) 3ln2=_______ |
Product rule
=2^3x+1*(1/5x-11)*5 + 3*ln2*2^3x+1 *ln(5x-11) ln8 = 2^3 |
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The natural log gives you the time needed to reach a certain level of growth.
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* e^x lets us plug in time and get growth.
* ln(x) lets us plug in growth and get the time it would take. |
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e represents the idea that all continually growing systems are scaled versions of a common rate.
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This is wild! e^x can mean two things:
* x is the number of times we multiply a growth rate: 100% growth for 3 years is e^3 * x is the growth rate itself: 300% growth for one year is e^3. Well, since the crystals start growing immediately, we want continuous growth. Our rate is 100% every 24 hours, so after 10 days we get: 300 * e^(1 * 10) = 6.6 million kg of our magic gem. |
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lne = ?
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1
* The math robot says: Because they are defined to be inverse functions, clearly ln(e) = 1 * The intuitive human: ln(e) is the amount of time it takes to get “e” units of growth (about 2.718). But e is the amount of growth after 1 unit of time, so ln(e) = 1. |
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Rate of change of a bubble
R=dV/dt=0.2 cm^3/sec and radius r=0.5 cmgiven differentiate sphere volume to get change with time: dV/dt V(sphere) =d/dt 4/3(pir^3)dr/dt |
V' = 4pir^2*r'
I want to know when r'=0.2 cm^3/sec set given rate equal to V' and solve for r' (rate of change of radius) given the radius (0.2cm^3/sec//4pi(0.5)^2 |
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Cost of production minimize ave cost
c(x) = x^3.... average cost(x) = c/x |
differentiate average cost
C' = x^2.... set equal to zero |
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Max profit
C = 150 + 40x x=80-price or price = 80-x Find maximum price/unit |
Profit = Revenue -Costs
R=price/unit*units = px C is given P = (80-x)x - (150+40x) P' = -2x + 20 set equal to zero to find maximum x=10 |
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Max profit truck rentals
costs 30 trucks at 20/day to run or 5/day storage |
Profit=Revenue - Costs = N(R-C)
N (#trucks) rented = 30 -(R-20(cost))=50-Rent P=(50-R)(R-C) C=5/day P=(50-R)(R-5) solve differentiate and set equal to zero |
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Race car: rate of change in distance bt it and spectator in final 100 feet if spectator is 200 ft from finish
A x y h 0 spectator car dx/dt is 176; 100ft out at x |
Pythagorean to find h as func x
differentiate to find changing h'dx/dt dx/dt is known, speed of auto. Plug in and solve. (-79 ft/sec) |
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y=cosx, x is given, dx/dt is given
find dy/dt in seconds at x |
differentiate y'=-sinx dx/dt
Plug in dx/dt value confused about how to solve |
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V shaped tank is
Vprism =1/2(xyL) (xy is fluid volume) (wh is tank dims given) V'-0.002 m3/s Find rate of vertical (x) decrease with time |
similar triangles can use proportions to write in other terms: y=wx/h so it is in x
solve differentiate for V in terms of x and dx/dt using given values of w and h, solve for dx/dt |
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implicit differentiation
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used when functions are in two variables when you cannot solve y in terms of x: solve for dx/dy and plug in values for x and y to find slope (instantaneous rate of change)
Instance of chain rule |
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Probability
Rolling sum greater than 7 or doubles |
Not mutually exclusive
Add together Pa + Pb - instances of doubles greater than seven |
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Postulates - given
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SSS
SAS ASA Reflexive leads to symmetric (another) leads to transitive (a third through a second) |
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Theorems - der from postulates
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AAS
HL Isosceles Triangle Parallel Lines (corr, alt in, alt out) Vertical angles congruent |
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Axioms -given
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Add, Multi, Div, Sub
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Why Reflexive?
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Thus, if perhaps two triangles share a side and you wish to prove those two triangles congruent using the SSS method, it is necessary to cite the reflexive property of segments to conclude that the shared side is equal in both triangles.
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Why Transitive?
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This holds true in geometry when dealing with segments, angles, and polygons as well. It is an important way to show equality.
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Other postulates
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midpoints, bisectors, defn of a line, supplementary 180 and complimentary 90 angles
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commutative
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add or mult or sub b from two equal lengths, get same amount.
vs transitive a=b, b=c, a=c. or a=c, b=d, a=b therefore c=d. |
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Additive equality
Additive inverse Additive identitiy |
AE= subtract four
AI= a+-a =0 cancels out 4 AIdentity= remove zero a+0=a |
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minimum even divisor for 72 and 42
Least common multiple |
factor, cross off only one similar in a pair between the two numbers. Multiply. 504
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Least common denominator
Larry 10 min/ele Moe 6 min/ele Curly x min/ele total together 3 min/ele |
to change to ele/min, invert.
Find least common Denominator, and reduce terms to solve for x |
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Inequality word problem
YMCA raffle wants >-32 K Cost of event 7250, P of ticket 25 How many need to sell? |
25t-720>=32K
t>=1570 |
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Bicycle Profits 3 speed vs 10 speed
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time cost equation for 480 minute work day
want profit per bike ,=300?? Max where x or y is zero. |
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Inflection point
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second derivative where slope = zero
a local max or min if first derivative on either side of point at zero is pos and neg (to show concavity changing) A) If on both sides of the point the f'slope is positive (or neg), then the inflection point is not a max or min. B)if on both sides of the point the f''slope is pos or neg, then the inflection point could be a local max/min (x^4) local max |
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Discern a functions max, mins concavity, and inflection points
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factor f' = 0
plug those numbers into original eq take second derivative to find if concave up or down. factor f'' set equal to zero to find inflection points |
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Maximize area if given circumference
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solve for one variable, substitute into Area equation, take derivative, set equal to zero and solve for variables.
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Area given, find dimensions for greatest volume
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isolate terms in Area to make single variable in volume. take derivative and solve for zero.
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Surface area of a cylinder without a top, find height and radius for max V
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SA isolate terms on either side of equation; plug into Volume. Derivative, set to zero, check the radius possibilities.
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Minimize distance from a graph of f(x) = sqrtx to the point (4,0)
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Pythagorean theorem
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binomial theorem
complex one real root two real roots |
-b+/-sqrt(b^2-4ac)/2a
b^2-4ac < 0 complex b^2-4ac = 0 one real b^2-4ac > 0 two real |
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Output maximum profit
50 apple tree = x 800 apples/tree Each tree loses 10 apples for each additional tree. What is maximum number of trees i can add before losses/tree take over? |
Output Total =
xtrees*Output/tree = 4000 Output/tree = 800 apples-10 apples *(x-50) =x( 800 apples -10 apples(x-50) Reduce, differentiate, solve for zero x=65 |
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Chain Rule
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Used when composite of two functions
not two separate functions (product or quotient) |
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synthetic sub
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bring down first coefficient of the dividend (not the divisor), mult by (x-2) then 2, add, continue to end.
Root if zero. Can Find other roots by factoring easier polynomial. Remainder/(x-2) ends new polynomial. |
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Integrate:
sinxdx cosxdx tanxdx |
sinxdx = -cosx
recall d/dx -cosx=sinx cosxdx = sinx recall d/dx sinx= cosx tanxdx = ln absvalue(secx) |
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Integrate
sin^2xdx cos^2dx |
sin^2x= x/2 - sin2x/4
cos^2x=x/2 + sin2x/4 |
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d/dx 2^(3x+1)
d/dx2^x |
Can't take derivative of base 2. Change to something we know how to derivate: e^x
d/dx 2^2x+1 = e^ln2^3x+1 = d/dx e^ln2*3x+1= 3ln2* d^ln2*3x+1 e^x^y = x^x*y In the same way: d/dx 2^x = d/dx e^ln2*x = ln2*e^lnx |
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d/dxlnx = d/dx log base e x =______
Find d/dx log base b x= ____________ |
1/x
Given b=e^lnb and b^k = x log base b (b^k) Substitute and take ln: ln( e^lnb*k)=lnx lnbk=knx k=lnx/lnb Differentiate lnb is a constant d/dx lnx = 1/x this 1/lnbx |
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Sequences are just progressions of numbers with a common difference separated by commas
Arithmetic sequence a(n) = |
a(n) = a(1) + (n-1)d
where d is the common difference (+/-) between arithmetic sequences |
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Sequences just numbers, numbers
with a common ratio separated by commas Geometric Sequence a(n) = |
a(n) = a(1) * r^n-1
where r is the common ratio between terms (*/divide) |
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Arithmetic sequences
1 5 9 13 17 |
a(n) = 1 + (n-1)4
common difference; only thing that changes is the starting point. Linear. |
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Geometric Sequences
2 6 18 54 3 1 1/3 1/9 1/27 |
a(n) = 2*3^(n-1)
a(n) = 3*(1/3)^(n-1) Common ratio structured so that for n=1, the ratio is r^0 or 1, which leaves the first term, or n=1. |
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Arithmetic Series progression of numbers with a common difference (+/-) separated by plus or minus
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The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
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Arithmetic Series
3 + 7 + 11 + 15 + ··· + 99 has a1 = 3 and d = 4. We solve 3 + (n – 1)·4 = 99 to get n = 25. |
Find a(n) using Arithmetic Sequence equation = a(1) + (n-1)d.
Multiply number of terms n, by the average of the first and last term. Series S(25) = 25(3+99/2) |
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Geometric Series progression of numbers with a common ratio between them and separated by a plus or minus
3 1 1/3 1/9 1/27 1/81 |
Sum of Series S(n) = a(1)(1-r^n)/(1-r)
S(6) = 3(1-(1/3)^6)/(1-(1/3)) ************************************ |
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Infinite series
convergent ratio divergent ratio |
ratios less than 1 converge
ratios greater or equal to 1 diverge |